Solutions of Unit 9 Chords of a Circle Step-by-Step
In this post, we provide Solutions of Unit 9 Chords of a Circle for Class 10 students. This unit is very important as it explains key concepts about chords and their properties in a circle. Understanding these concepts will help you solve many geometry problems related to circles.
The chapter covers some important theorems, such as:
- A unique circle can pass through three non-collinear points.
- The perpendicular drawn from the center of a circle always bisects the chord.
- Congruent chords are always at the same distance from the center of a circle.
These theorems are explained with examples to make them easy to understand. They are very helpful for solving questions in your exams.
Our solutions are written step by step so that you can follow them easily. Each solution is explained in simple words to help you learn faster. If you read this post carefully, you will improve your understanding of geometry and do well in this topic.
PDF Solutions of Unit 9 Chords of a Circle
By studying these Solutions of Unit 9 Chords of a Circle, you will be ready to solve any questions about chords and circles confidently.
Solutions of Unit 9 Chords of a Circle Key Points
One and Only One Circle Through Three Non-Collinear Points:
- Proof: In geometry, any three points that do not lie on the same straight line (i.e., non-collinear points) can uniquely determine a circle. This is because there is only one possible way to draw a circle that will pass through all three points. The center of this circle can be found by locating the perpendicular bisectors of any two line segments connecting these points. Where these bisectors intersect is the circle’s center, and the distance from this center to any of the three points becomes the radius. This property is fundamental in circle geometry because it establishes that three points are sufficient to uniquely define a circle.
- Corollary: Given any three non-collinear points, there is exactly one circle that can be drawn through them. This means that if you have three points arranged such that they do not fall in a straight line, you can always construct a single, unique circle that passes through each point. This corollary is often used in constructions and proofs involving circles, as it provides a foundation for understanding the relationship between points and circles in a plane.
Perpendicular from the Center Bisects a Chord:
- Proof: A chord is a line segment that connects two points on the circumference of a circle. When you draw a line from the center of the circle to the midpoint of this chord, and this line is perpendicular to the chord, it will divide the chord into two equal halves. This is because, in a circle, the line drawn from the center to a chord is always perpendicular when it bisects the chord. This theorem is useful in understanding symmetry within circles and is often applied in problems where distances and symmetrical properties of circles are considered.
- Corollary: The perpendicular line drawn from the center of the circle to a chord will always divide the chord into two equal parts. This means that if a line from the circle’s center crosses a chord at a 90-degree angle, the chord’s length on either side of this intersection will be the same. This corollary is frequently applied in geometric proofs and constructions to establish equal lengths in symmetrical shapes involving circles.
Chord Equidistant from the Center:
- Proof: If two chords in a circle are equal in length (congruent), they will be at the same distance from the center of the circle. This can be visualized by imagining two equal-length chords in a circle; because of their equal lengths, the perpendicular distance from the center of the circle to each chord will be the same. This property highlights an important aspect of symmetry in circles, where congruent elements are consistently spaced in relation to the center.
- Corollary: Two chords that are equidistant from the center of a circle are congruent, meaning they have the same length. If you measure the perpendicular distance from the center to two different chords and find it to be the same for both, you can conclude that these chords are equal in length. This corollary is particularly useful in solving geometric problems that involve identifying equal lengths and distances within circles, as it provides a reliable method for establishing chord congruence based on their distance from the center.
Each of these theorems, along with their proofs and corollaries, forms an essential part of understanding chord properties in circles. They offer foundational principles used widely in geometry to solve problems, conduct proofs, and understand symmetrical relationships within circular shapes.
Multiple Choice Questions Test for Unit 9
To further solidify your understanding of solutions of Unit 9 Chords of a Circle, we’ve included a Multiple Choice Questions (MCQ) test covering the unit’s key concepts. This test will reinforce the proofs and corollaries related to unique circles through three points, perpendicular bisectors of chords, and the relationship between equidistant chords and their congruency. Taking this MCQ test will help you review the core principles and prepare for exam questions on chords in a circle.
Mastering these Solutions of Unit 9 Chords of a Circle gives you a solid foundation in circle geometry, particularly in understanding chords and their properties. By working through these solutions and revisiting key points, you’ll feel more confident in approaching related problems in exams. Practice regularly and refer to additional resources if needed to keep reinforcing your skills in geometry. With these step-by-step solutions, you’re well-equipped to handle any challenges involving chords of a circle. Lets move on to the Solutions of Unit 10. If you want to revisit the previous concepts do check Solutions of Unit 8. Video Play List of all lectures of solutions of unit 9 chords of a circle.